In Defense of Memorizing Procedures

A common goal of inquiry is that the students will invent or discover scientific truths for themselves. In particular, they will invent, not memorize, procedures. In one class early in the semester, my students invented complicated equations of motion — something they had never achieved before in my class. Yet they achieved this by first memorizing some basic procedures.

A few of the procedures they needed were:

1. To deduce the displacement from a “velocity vs. time” graph, find the area under the curve or line of the graph.

2. To find the area of a rectangle, find its base times its height. To find the area of a triangle, use one half of its base times its height.

3. To find the change in velocity (△v), multiply acceleration (a) by the change in time (△t).

Students can derive an equation of motion from these procedures and the following diagram:

To find the right equation, they have to first find the area of the triangle, then the rectangle, then add them together.

If they also use fact #3, they can find a second equation. It is actually possible for students to find many different equations that describe the diagram, such as:

In my earlier days as an education researcher, I would have been dissatisfied with this approach. For one thing, I had to tell students the three facts, and I had to draw the picture for them. We had to practice finding areas of simpler figures first. My past self would have preferred to make them “invent” all of the facts, procedures, and diagrams. But I now believe that this approach to inquiry doesn’t always work. “Pure” inquiry can be a cheat: it leverages the bits of grade school mathematics and physics knowledge that students are fortunate enough to remember. Therefore, it successfully serves students who recently took a math course and excelled in it, such as most physics majors, but punishes those who did not.

I suggest that you, the reader, might please take a moment to think about a topic that you learned three or more years ago with no refreshers since then. (Not something that is of professional importance to you.) How well do you remember it? In college, I remember taking a particular history course that I enjoyed a great deal. Today, I remember almost nothing from it. Why, then, should I expect my students to recall that “A = (1/2) b h”?

My belief is that inquiry doesn’t work unless it first levels the playing field; students should be provided with the basic processes they need to work with in order to solve the problem.

The education researcher Anna Sfard [1] points out that when children learn to count, there is a stage when they will say “one, two, three” to count three marbles, but will not articulate that there are “three” marbles. Instead, they will say that there are “one, two, three” marbles. This child has internalized the process of counting, but does not yet perceive the number “three” as an object independent of this process. Sfard argues that this pattern of “process before object” is universal to mathematics. A fraction “was initially regarded as a short description of a measuring process rather than as a number.” To understand a fraction as a kind of number is to “reify” it.

Sfard’s asks whether “when a person gets acquainted with a new mathematical notion, the [procedural] conception is usually the first to develop?” She answers this question in the affirmative.

From this perspective, it seems unreasonable to expect student to “reify” the velocity diagram and to understand it as an instance of an equation, without extensive experience with various processes. These processes might include (1) using the area to find displacement, (2) finding areas of rectangles and triangles, (3) using “rise over run” to relate velocity to acceleration.

In my opinion, it is simply impossible for students, in my one semester algebra-based physics course, to derive all of these processes by the inquiry method. Therefore, my method includes a great deal of procedural guidance. But I don’t view this as a defeat! I am happy to provide students with exactly as much process as is appropriate — no more and no less. Finding that balance is part of the art of teaching, and it must be found separately in every different subject matter and for every group of students.

(1) Anna Sfard, “On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin,” Educational Studies in Mathematics, Vol. 22, №1 (Feb. 1991), pp. 1–36.

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